Find the Rank [[1,-4,2],[-2,8,-9],[-1,7,0]]

Problem

rank(1)

Solution

1. Set up the matrix A to perform row reduction to row echelon form.

A=[[1,−4,2],[−2,8,−9],[−1,7,0]]

2. Eliminate the first entries of the second and third rows using the first row.

(R_2)→(R_2)+2*(R_1)

(R_3)→(R_3)+(R_1)

[[1,−4,2],[0,0,−5],[0,3,2]]

3. Swap the second and third rows to organize the pivot positions.

(R_2)↔(R_3)

[[1,−4,2],[0,3,2],[0,0,−5]]

4. Identify the number of non-zero rows in the row echelon form.

Row 1: *[1,−4,2]

Row 2: *[0,3,2]

Row 3: *[0,0,−5]

5. Determine the rank based on the count of linearly independent rows.

Number of non-zero rows=3

Final Answer

rank(1)=3

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